If you prefer a smaller park with easy access the Douglas Apsley National Park on the east coast will appeal. A bit over 2 hours from Launceston or Hobart this park has day visitor facilities and camping facilities with more basic camping areas further into the park. Short and easy walks abound but there are overnight walks as well. Some of the waterfalls are a sight to behold.
If you're an experienced trekker, the Leeaberra Walk is a 2-3 day highly recommended experience. This is not a wilderness area like elsewhere although there are some untouched areas within the park.
But it does have several wonderful eucalyptus areas and rare and endangered animals. It doesn't get the publicity of some other parks but is still well worth a visit.